Options in managerial compensation

(1)

INSTITUTO SUPERIOR DE CIÊNCIAS DO TRABALHO E DA EMPRESA FACULDADE DE CIÊNCIAS DA UNIVERSIDADE DE LISBOA

DEPARTAMENTO DE FINANÇAS DEPARTAMENTO DE MATEMÁTICA

Options in Managerial Compensation

Carla da Costa Tavares

(2)

(3)

Acknowledgements

First, I would like to thank my supervisor, Professor José Carlos Gonçalves Dias, for his guidance and support throughout my dissertation.

A very special gratitude to my family. I have no words to express how your support was meaningfull in this journey.

I would also like to thank Breno Gonçalves for all the support given along the way. You have been a crucial encouragement during the completion of this thesis and a true love along this road.

Finally, I would also like to thank all of my friends, specially to Ana Gameiro for the friendship and companionship provided during the master’s degree, to Mário Gamero and to Diogo Cardoso for the support provided when I needed the most.

This accomplishment would not have been possible without you all.

(4)

Resumo

O trabalho desenvolvido nesta tese baseia-se no artigo de Ju, N. & Leland, H. & Senbet, L.W. (2014) [8] que examina os efeitos das opções sobre acções incluídas nos pacotes de compensação de executivos nas políticas de risco de investimento. Começamos por introduzir opções standard para compreender melhor como as opções sobre acções são importantes componentes destes pacotes de compensação. Como vamos poder observar, opções de compra aumentam de valor quando o preço do activo subjacente aumenta, e descem no caso contrário. Desta forma, as opções sobre acções fazem a ligação entre a performance das acções e a compensação de executivos.

Apesar disto, as opções de compra são também uma função crescente de volatilidade e poderíamos argumentar que isso poderia encorajar os gestores a optar por escolhas de risco em excesso. No entanto, também temos que ter em conta a aversão ao risco do gestor: um gestor averso ao risco pode preferir abdicar de receber um montante superior por uma incerteza menor no futuro.

Tendo em conta um executivo averso ao risco, um contrato típico de compensação com opções de compra fornece uma escolha de risco de investimento sub-óptima. Incluir mais opções de compra no pacote de compensação também induz a escolhas de risco inferiores uma vez que, mesmo que o payoff seja superior, a sua volatilidade também aumenta.

Assim, é improvável que esta distorção seja corrigida apenas com opções standard nos pacotes de compensação empresarial.

Nesta tese vamos examinar a escolha de risco de investimento e qual o pacote óptimo de compensação empresarial quando este é composto por pagamentos fixos, acções e diferentes tipos de opções sobre acções: opções de compra, opções de venda e lookbacks.

A nossa perspectiva é a de minimizar o custo total da firma ao fazer alterações nas componentes do portfolio de compensação do gestor.

Vamos também analisar as diferenças obtidas quando fazemos alterações em alguns parâmet-ros e vamos ter em conta que o gestor faz as suas escolhas com vista a maximizar a sua utilidade esperada.

Existem enúmeras literaturas acerca dos custos da firma associados ao conflito existente entre os vários associados da mesma. Parrino e Weisbach (1999) [11] estudaram a mag-nitude do conflito entre accionistas e obrigacionistas usando a simulação de Monte Carlo. No nosso caso, e para simplificar o modelo, vamos apenas ter em conta o conflito entre o gestor e o accionista (mesmo sabendo que o comportamento do gestor também influencia o

(5)

obrigacionista).

Também a ter em conta no nosso modelo, estamos a assumir que a mudança de nível de risco também afecta o valor da firma. Ross (2004) [12] estuda os efeitos dos incentivos em opções de compra e opções de venda mas, tal como noutras investigações, assume que a escolha do risco não afecta o valor da firma. Nós estamos particularmente interessados nas políticas de investimento que não só afectam o nível de risco, mas também afectam o valor da firma.

Como principal objectivo deste trabalho estamos preocupados com o potencial conflito entre o gestor e os accionistas e como é que esta relação é afectada pelas escolhas de risco de investimento induzidas pelas compensações de opções.

A tese está organizada da maneira a seguir apresentada: no Capítulo 1 introduzimos conceitos básicos de opções, seguidos pela definição e avaliação de opções standard e lookback. Em seguida, no Capítulo 2, fazemos a caracterização do problema e classificamos as componentes dos pacotes de compensação empresarial e do risco associado. Depois da ap-resentação do modelo, analisamos e apresentamos os resultados obtidos no Capítulo 3. As soluções analíticas foram implementadas em Matlab.

(6)

Abstract

In this thesis, we will present the effects of options in managerial compensation.

Stock options are frequently used in these compensation packages and they can provide desirable incentives to the executives, but they can also have adverse risks and distort the choice of investment risk. This choice is dependant on the risk aversion that the manager might have and the firm itself.

In order to correct this distortion we will combine different types of components such as fixed payments, stocks and different types of stock options: calls, puts or lookbacks. To accomplish our goal we take in mind that the manager will want to maximize her expected utility. The analytical solutions proposed are implemented in Matlab.

Finnally, we expose results for each managerial compensation package studied and some conclusions are presented: we show that the inclusion of lookback call options have positive effects in the choice of risk policies and can induce risk policies that increase shareholder wealth.

Keywords: Managerial Compensation, Option Pricing, Black-Scholes-Merton Model,

(7)

List of Tables

3.1 Risk effects of compensation contracts with regular calls. . . 15 3.2 Number of regular calls and company shares that performed the same

man-ager’s utility as Table 3.1. . . 18 3.3 Number of regular calls and puts that performed the same manager’s utility

as Table 3.1. . . 19 3.4 Number of lookback calls that performed the same manager’s utility as Table

3.1. . . 21 3.5 Number of company shares and lookback calls that performed the same

(10)

1 Introduction

In this thesis we will examine the effects of stock options envolved in the managerial compensation on corporate investment risk policies. We start to introduce standard options in order to get a better understanding of executive stock options because they are an important component of managerial compensation. As we will notice, calls increase their value when the underlying asset is going up and decrease otherwise. This way, these options connect the stock performance with the manager’s compensation.

Despite this, call options are also an increasing function of volatility and we can also argue this might encourage managers to undertake risky investments in excess. But we also need to take in mind the managerial risk-aversion: a risk-averse manager may prefer giving up a higher current stock price for a lower future uncertainty.

Taking a risk-aversion manager, a typical compensation contract with call options provides a sub-optimal level of risk investment. Taking more calls in the compensation contract will also induce lower level risks - even though we will have a higher payoff, the volatility will be higher too.

Therefore, it is unlikely to correct this distortion with simple options in the managerial compensation.

We will examine the choice of risk and the optimal managerial compensation when the compensation package is composed by fixed payments, stocks and different types of stock options: calls, puts or lookbacks.

Our perspective is to minimize the total cost to the firm by changing the components of company stock and options in the manager’s portfolio.

We also analyse the differences through the changes of some parametric values and take in mind that the manager chooses the risk level in order to maximize expected utility.

There are enumerous literature about the agency costs associated with the conflict of interests among the firm’s various claimants.

Parrino and Weisbach (1999) [11] studied the magnitude of the conflict between stock-holders and bondstock-holders using Monte Carlo simulation.

(11)

In our case, in order to simplify the model, we will only take in mind the conflict be-tween the manager and the stockholder (even that manager’s behaviour also interferes with bondholders).

Also, in our model, the changing of risk level may also affect the firm’s value. Ross (2004) [12] studies the risk incentives effects of puts and calls, but as in other papers, he considers that the choice of risk level does not affect firm’s value. We are particularly interested in the investment policies that not only affect the risk level but also the value of the firm.

As the main driver for this thesis we are concerned with the potential conflict between the manager and the shareholders and how this is affected by corporate investment risk induced by option-type compensations.

This thesis is organized as follows: in the Chapter 1 we introduce options basic concepts, followed the definitions and evaluation of standard and lookback options.

Proceeding, in the Chapter 2 we denote our problem and classify the components of managerial compensation and corporate investment risk. After the presentation of the model, we analyse and present the numerical results obtained in the Chapter 3. The analytical solutions proposed are implemented in Matlab.

(12)

1.1 Building Blocks

1.1.1 Options Basic Concepts

An option is a contract in which their holders have the right to buy or sell an asset, but not the obligation to do so, unlike other contracts such as futures or forwards. The most basic options are calls and puts.

A call option gives the option holder the right, but not the obligation, to buy a specific amount of an asset at or until a determined time in the future, called the ’expiration date’ or ’maturity date’, for a pre-specified price, defined as the ’strike price’. On the contrary, a put option gives the option holder the right, but not the obligation, to sell the asset.

Options can also be divided into two categories in terms of the time at which they can be exercised: European or American options. European options can only be exercised at the maturity date T, while American options can be exercised at any time until T.

Calls increase their value when the underlying security is going up, and decrease in value when the underlying asset is going down. On the other hand, puts increase in value when the underlying asset declines in price and decrease in value when it is going up.

1.1.2 Geometric Brownian Motion Assumption and the

Black-Scholes-Merton Model

In order to require a price to be quoted in the market for these kinds of financial products, Fisher Black, Myron Scholes and Robert Merton developed a formula of greater importance. The Black-Scholes-Merton (BSM) Model, introduced in 1973 ([2] and [10]), is an option pricing model based on the Geometric Brownian Motion (GBM).

A GBM is a continuous-time stochastic process that satisfies the following stochastic differential equation:

dSt

St

= µdt + σdWt, (1.1)

where we have a given moment in time t, the asset’s price St, the drift µ (which measures the annualized average return) and the volatility σ (which corresponds to the annualized standard deviation of the underlying asset price return). Note that the coefficients µ and

σ are both constants in this model. {Wt; t ≥ 0} is a Brownian motion, which has a normal distribution with mean 0 and variance t, under the observed probability measure.

(13)

We use the GBM assumption because it only assumes positive values, just like stock prices, and it is simple to implement.

The BSM model is used to calculate the theoretical value of a European-style option using current stock prices, expected dividends, the option’s strike price, expected interest rates, time to maturity and expected volatility.

Although the BSM Model revolutionized the financial markets, its underlying assumptions are not observed in the market:

i) The underlying asset price follows a GBM process - the above solution St (for any value of t) is a log-normally distributed random variable;

ii) The risk free rate r, and the volatility σ, are known constants over the option’s life; iii) There are no transactions costs associated with hedging a portfolio;

iv) There are no arbitrage opportunities - all risk-free portfolios must earn the same return; v) Trading takes place continuously in time;

vi) Short selling is permitted and the assets are perfectly divisible.

The inverse relation between the strike price and the implied volatility, known as volatility

smile is not captured by this model. This way, other models were studied.

The Constant Elasticity of Variance (CEV) model, which was developed by Cox (1975) [4], is one model that is consistent with the existence of a negative correlation between stock returns and realized volatility, known as leverage effect (see Bekaert and Wu, 2000) [1], and with the volatility smile (see Dennis and Mayhew, 2002) [5].

The comparison between these two models was done by enumerous authors, specific by MacBeth and Merville (1980) [9] that concluded that prices obtained with CEV model were closer to market values especially when the elasticity parameter was negative. Also Boyle and Tian (1999) [3] concluded that the difference between the two models was higher when calculating the price for path-dependent options than for standard options.

Concluding, we also need to take in mind that as closer as we get to reality, harder and more complex the model will become. In our case we focus the attention to the simplest version of the BSM model with the assumptions defined earlier.

(14)

1.1.3 Standard Options

For the solutions of BSM model we will use the following notation:

• V (S, t) is the function that gives us the value of an option (C(S, t) for a call and

P (S, t) for a put) depending on the current value of the underlying asset, S, and time t (0 ≤ t ≤ T, where T is the maturity date);

• Stis the price of the underlying asset at time t; • K is the strike price or exercise price of the option; • σ is the volatility of the underlying asset;

• r is the risk free interest rate; • q is the dividend yield.

The Black-Scholes partial differential equation is given by:

∂V ∂t + 1 2σ 2_S2∂2V ∂S2 + (r − q)S ∂V ∂S − rV = 0 (1.2)

Definition 1. At t = T , the value of the call is known with certainty to be the terminal

payoff:

C(S, T ) = max{ST − K, 0} = (ST − K)+ (1.3)

Definition 2. At t = T , the value of the put is known with certainty to be the terminal

payoff:

P (S, T ) = max{K − ST, 0} = (K − ST)+ (1.4) Both give us a final condition of the problem. This way, for each type of option, we will have a unique solution to the aforementioned partial differential equation.

The Black-Scholes-Merton exact solutions of the European call and put options are, re-spectively, given by:

Ct(S, K, T ) = Ste−q(T −t)Φ(d1) − Ke−r(T −t)Φ(d2), (1.5)

Pt(S, K, T ) = Ke−r(T −t)Φ(−d2) − Ste−q(T −t)Φ(−d1), (1.6)

(15)

Φ(x) = √1 2π Z x −∞ e−t22_dt, and d1= ln(St/K) + (r − q +1₂σ2)(T − t) σ√T − t , and d2= d1− σ √ T − t.

(16)

1.1.4 Lookback Options

The European lookback options were first described by Goldman, Sosin and Gatto (1979) [7]. They are exotic options whose terminal payoff depends on the maximum or minimum value of the underlying asset price during the option’s life. This allows their holders to speculate on a market upward or downward move, with the advantage of not knowing the exact date of its occurrence (during the option’s life).

There are two types of lookback options: floating and fixed strike lookback options. In this thesis we are only detailing the floating strike lookback options.

A floating strike lookback call gives the option holder the right to buy at the lowest price recorded during the option’s life. A floating strike lookback put gives the holder the right to sell at the highest price recorded during the option’s life.

Therefore, these contracts are no-regret options since they will never finish out-of-the-money - in a call option the strike price will never be higher than the market price of the underlying asset, and in a put option the strike price could not be lower than the market price of the underlying asset.

In the following definitions, ST represents the asset price at option’s maturity T, while

m0,T and M0,T denote the minimum and maximum asset prices recorded during the option’s

life, respectively.

Definition 1. The time-T value of a floating strike lookback call option on the asset S,

with a unit contract size, inception at time 0 and expiry date at time T (≥0) is:

LCf l(T ; ST; m0,T; T ) = ST − m0,T (1.7)

Definition 2. The time-T value of a floating strike lookback put option on the asset S,

with a unit contract size, inception at time 0 and expiry date at time T (≥0) is:

LPf l(T ; ST; M0,T; T ) = M0,T− ST (1.8) At any valuation date t, 0 ≤ t ≤ T, the fair value of the floating strike lookback call and the floating strike lookback put are given by the following risk-neutral expectations:

LCf l(t; St; m0,t; T ) = e−rτEQ[ST− m0,T | Ft] (1.9) and

(17)

It will be necessary to derive probability density fuctions for the maximum and for the minimum of the underlying spot price. To compute the price of a lookback option under the Black-Scholes-Merton model we can also use the pricing formulae of Domingues, A. thesis [6] based on Zhang (1998, p.341-352) [13].

The prices of a floating strike lookback call and a floating strike lookback put, at 0 ≤ t ≤ T and during the option’s life, are given by:

LCf l(t; St; m0,t; T ) = =                        Ste−qτΦ[dbs1(St, m0,t)] − m0,te−rτΦ[dbs(St, m0,t)] +St δ e−rτ St m0,t −δ Φ[dbs(m0,t, St)] − e−qτΦ[−dbs1(St, m0,t)] , r 6= q Ste−qτΦ[dbs1(St, m0,t)] − m0,te−rτΦ[dbs(St, m0,t)] +Ste−rτσ √ τ f [−dbs1(St, m0,t)] −Ste−rτ σ 2 2Φ[−dbs1(St, m0,t)] n 2 σ2 ln St m0,t + τo, r = q (1.11) LPf l(t; St; M0,t; T ) = =                        M0,te−rτΦ[−dbs(St, M0,t)] − Ste−qτΦ[−ddbs1(St, M0,t)] +St δ e−qτΦ[dbs1(St, M0,t)] − e−rτ St M0,t −δ Φ[−dbs(M0,t, St)] , r 6= q M0,te−rτΦ[−dbs(St, M0,t)] − Ste−qτΦ[−ddbs1(St, M0,t)] +Ste−rτσ √ τ f [dbs1(St, M0,t)] +Ste−rτ σ 2 2 Φ[dbs1(St, M0,t)] n 2 σ2ln St M0,t + τo, r = q (1.12)

where Φ[.] is the normal cumulative distribution function, f [.] is the normal probability density function,

(18)

and where all contracts are initiated at time zero, m0,t and M0,t are the minimum and

maximum prices recorded until date t, St is the underlying price at time t, and τ = T − t is the time remaining to expiration.

(19)

2 Managerial compensation and corporate

investment risk

In this section we will show how alternative mixtures of company stock, stock options and cash and stock holdings in other companies affect the investment risk policy. We take into account that the effects of executive stock option choices are related to the risk level, which in turn affects the firm value. In order to do so, this chapter follows the highlights of Ju et al. (2014) [8].

2.1 Firm value as a funtion of risk

We assume that the initial value of the firm is given by:

V0(σ) = V0− a

 σ − σ0

σ0

2

, a ≥ 0 (2.1)

V0 is the optimal firm value and a is a constant that measures the cost associated to the

deviation from the optimal volatility level, σ0. V0 also represents an "efficient frontier" with

regard to the value and risk of the firm. In other words, it represents the highest firm value among all the same firm risk strategies. This fuction has a maximum value V0 at the risk

(20)

when it is less than 1.

2.2 Expected return as a function of risk

The value of the firm evolves, for each volatily level (σ), according to the following diffusion process:

dVt(σ)

Vt(σ)

= µV(σ)dt + σdBVt , (2.3)

where {BtV; t ≥ 0} is a standard Brownian motion.

In order to simplify, we wil specify the value of the assets in other companies as another diffusion process, given by:

dSt

St

= µSdt + σSdBtS, (2.4)

where {BSt; t ≥ 0} is another standard Brownian motion. We will also denote ρ as the correlation between {BV

t ; t ≥ 0} and {BtS; t ≥ 0}.

In the Black-Scholes-Merton model, as the options can be easily covered and changed they can be priced assuming the risk-free rate to their return. However, as executives are usually not allowed to sell or hedge their stock options, we will have to take into account their subjective return distribution to determine their expected utilities. According to this, we assume: µV(σ) = r + σ σ0 (µV(σ0) − r) ⇔ µV(σ) − r µV(σ0) − r = σ σ0 , (2.5)

where µV is the expected return corresponding to risk level σ and r is the risk-free rate. Folllowing the Capital Asset Pricing Model (CAPM),

µV(σ) − r

µV(σ0) − r

= Cov(˜µV(σ), ˜µm)

Cov(˜µV(σ0), ˜µm)

, (2.6)

where ˜µV(σ) is the return corresponding to risk level σ and ˜µm the return of the market. The CAPM assumes that the agents are price takers, mean-variance optimizers and have homogeneous expectactions. That is, they have the same way to analyse securities and, as a result of that, the same beliefs about future returns. As a theory of financial equilibrium, it also makes the assumption that the currently observed asset prices are in equilibrium.

(21)

If we assume that the covariance is proportional to the risk level, σ, Cov(˜µV(σ), ˜µm) Cov(˜µV(σ0), ˜µm) = σ σ0 , (2.7)

then we obtain Eq. (2.5).

2.3 Executive’s terminal wealth and optimal risk policy

The value of the firm and the value of the assets in other companies at time T (terminal values) are given by:

VT = V0(σ)e(µV(σ)−σ 2_{/2)T +σB}V T_, _(2.8) and ST = S0(σ)e(µS−σ 2 S/2)T +σSBST_. _(2.9)

To determine the terminal wealth we will assume that the executive has the following assets:

F - a risk-free investment/manager’s non-company wealth invested in the riskless asset; S0 - value of shares of other companies;

NS - shares of company stock V0(σ);

NC - call options with strike K and T years of maturity in the portfolio.

In order to determine the terminal value of the executive’s wealth, we have to take in mind the value of the company holdings (shares and call options). Therefore we have two possibilities: If VT(σ) < K, then the options will be out of the money, and the value of their holdings is given by NSVT(σ).

Alternatively, if VT(σ) > K, the manager will exercise the call options, paying the strike price K. This in turn will dilute the original shares by the factor 1 + NC.

(22)

The manager makes her choices in order to maximize her expected utility. We will also assume that the manager as a constant relative risk-aversion utility fuction, where relative risk aversion is the same at every wealth level:

U (WT) =

W_T1−Λ

1 − Λ , Λ ≥ 0, Λ 6= 1, (2.11)

where Λ is the coefficient of relative risk-aversion of this utility function. An investor with a higher Λ, will be more risk averse.

The manager will choose the volatility level by maximizing her expected utility. According to Eqs. (2.10) and (2.11): max σ E [U (WT)] = max σ E " W_T1−Λ 1 − Λ # = max σ E      F erT _{+ S} T + NSVT(σ) + NC(1 − NS) 1 + NC max(VT(σ) − K, 0) 1 − Λ 1−Λ     (2.12)

(23)

3 Numerical Results

In this chapter, we will present and analyse the numerical results. Since equation (2.12) cannot be solved analytically, we will obtain the corresponding numerical solution of the maximization problem.

3.1 Risk effects of compensation contracts with regular

calls

In this section, we will denote N CW0 as the initial non-company wealth and fN C as the fraction of N CW0 invested in other companies. This way, the investment in the riskless

asset is F = (1 − fN C)N CW0 and the number of shares in other companies is given by

S0= fN CN CW0.

In the simulation results we use the following base values: a = 50, V0 = 100, r = 5%,

σ0 = 0.38, µV(σ0) − r = 7%, σS = 0.2%, ρ = 0.2, µS = 12%, Λ = 2, N CW0 = 0.32,

fN C = 0.8, T = 5, NS = 0.32% and NC = 0.38%. These values were based in a typical market and historical averages.

We will study the changes and effects of risk-aversion coefficient (Λ), the portion of call options, company shares and non-company shares (NC, NS, fN C), the option strike price (K) and diversification of components which is given by N CW0.

(24)

We can also find the set of parameteres that combined result in the choice of the optimal volatility level, σ0= 0.38, for a particular Λ.

The full results are shown in the next Table:

Table 3.1: Risk effects of compensation contracts with regular calls.

σ V0(σ) VC TC E [U (WT)] 103 dE[U ] dV 10 3_PPS Base 0.271 95.898 32.567 4.532 -0.981 4.703 4.886 Λ = 0 0.553 89.599 47.529 10.868 1.637 12.148 12.148 Λ = 4 0.175 85.432 23.512 14.930 -0.514 7.891 4.425 a = 10 0.184 97.337 27.345 3.078 -0.953 4.867 5.364 a = 30 0.239 95.894 30.475 4.528 -0.971 4.796 5.088 a = 70 0.290 96.108 33.918 4.328 -0.987 4.634 4.753 a = 90 0.304 96.393 34.924 4.047 -0.992 4.580 4.655 NC= 0.0% 0.290 97.201 34.283 3.110 -1.095 4.863 4.055 NC= 0.2% 0.280 96.548 33.388 3.828 -1.027 4.774 4.524 NC= 0.5% 0.264 95.330 31.894 5.134 -0.955 4.658 5.104 NC= 1.0% 0.239 93.150 29.600 7.440 -0.873 4.560 5.979 NS= 0.0% 0.347 99.630 39.097 0.518 -1.576 2.414 0.972 NS= 0.2% 0.283 96.758 33.667 3.563 -1.140 4.467 3.434 NS= 0.5% 0.259 94.933 31.445 5.660 -0.813 4.673 7.073 NS= 1.0% 0.244 93.621 30.064 7.428 -0.553 4.026 13.148 fN C = 0.0 0.275 96.155 32.884 4.277 -1.065 5.386 4.744 fN C = 0.5 0.274 96.097 32.812 4.334 -1.002 4.822 4.807 fN C = 1.0 0.267 95.584 32.191 4.843 -0.973 4.703 4.967 K = 0.5V0(σ) 0.236 92.847 57.134 7.666 -0.864 4.905 6.577 K = 0.8V0(σ) 0.249 94.092 39.930 6.360 -0.942 5.012 5.651 K = 1.2V0(σ) 0.283 96.761 26.557 3.649 -1.010 4.663 4.571 K = 1.5V0(σ) 0.295 97.517 19.877 2.870 -1.039 4.605 4.266 N CW0 = 0.2 0.250 94.147 30.601 6.269 -1.236 7.327 4.797 N CW0 = 0.5 0.292 97.346 34.494 3.096 -0.756 2.884 5.050 N CW0 = 1.0 0.332 99.200 37.870 1.260 -0.468 1.195 5.448

3.1.2 Effect of investment technology

The parameter a reveals the costliness of deviating from the optimal volatility level (σ0):

a smaller a reprensents that is less costly to deviate.

In our results, the deviation of the firm value, V0(σ), from V0(σ0) = V0is most expressive

for moderate values of a.

(25)

from the optimal level as possible, because any deviation will lower the initial firm value. Also, in the base parameters, the investor is risk-averse, so will have a tendency to choose a lower risk level.

At the same time, the manager will also want to increase the value of the call, by increasing the risk-level.

The final choice of the investor will be the predominant factor on all the aboves. In our case, the risk-aversion factor is the dominant one as the risk level choosen in all the cases is not either near or above the optimal value, σ0= 0.38, but below.

3.1.3 Effect of increasing the portion of call options, company shares or

non-company-shares

Regarding the effect of increasing the portion of call options (NC), company shares (NS) or non-company shares (fN C) the intuition is that the risk level will also increase. However, this is only valid for a risk-neutral manager.

As in our results the manager is risk averse, the manager will want to adopt safer invest-ments and lower the risk level of the firm to reduce the portfolio risk. This is a consistent result for call options, company shares and non-company shares: with the increase of the portion of these components in the manager’s portfolio, it becomes riskier and the manager will want to reduce the portfolio risk.

3.1.4 Effect of option strike price

As we can see in Table 3.1, the resulting risk levels are many times below from the optimal risk level, σ0 = 0.38. We will now consider the effect that strike price has in the incentive.

As expected, the higher the strike price is, the higher the risk level the manager is going to take.

(26)

possibility for the firm value, but is the maximizing one. The manager may adopt other possibility that can result in a lower firm value for the same risk level.

3.1.6 Utility and pay-performance sensitivities with respect to the firm

value

In all the tables shown in this thesis we represent in the last columns the partial derivative of the expected utility with respect to the initial firm value, dE[U ]_dV , and the partial derivative of the manager’s certainty equivalent with respect to the initial firm value, dU−1_dV(E[U ]). These are also known as utility sensitivity and pay-performance sensitivity (PPS), respectively.

The PPS is the relationship between executive’s total pay and performance. It measures the change in the manager’s payoff that is associated to a given performance of the company that she leads.

Ceteris paribus, a larger sensitivity is a sign of a better alignment between the executive

incentives and the performance of the company. The sensitivity is larger when the pay responds more to changes in performance.

We could maximize these indicators but they do not measure the magnitude of agency cost resulting from suboptimal choice of the risk level as they are only dependent on the value and structure of compensation package. In line with this, we choose to minimize the total cost of the firm and keep the same utility level of the Table 3.1.

3.1.7 Minimizing the total cost of the firm

After evaluating the risk effects of compensation contracts with regular calls, we will now consider combinations of other stock-based components. The best combinations will minimize the total cost to the firm and assume the same utility level as the manager would have if choose the portfolio correspondent from each entry in Table 3.1.

The total cost is defined as the agency cost, (cost of deviating from the optimal firm value V0(σ0)) plus the market values of the company shares and regular calls/stock-based

components in the compensation.

The total cost will not include the assets held in other companies because the firm does not control them. Company shares are the first stock-based component that we will analyse and the results are revealed in the next Table.

Table 3.2 shows that for most of the cases, the total cost is lower than the total cost of Table 3.1 - achieving the same utility level is less costly to use company shares instead of regular options. This happens because a risk averse manager assumes options as the most risky assets in the portfolio. This brings us to the question whether is worth to have options

(27)

in the compensation package. In the situation where the manager holds a substantial non-company wealth (N CW0) call options may become desirable. As we can see in the last entry

of Table 3.2, the managers should increase their option holdings: with an augmentation on call options from 0.38% (base value in Table 3.1) to 0.6%.

Table 3.2: Number of regular calls and company shares that performed the same manager’s utility as Table 3.1. σ V0(σ) 102NS VC 102NC TC 103 dE[U ] dV 10 3_PPS Base 0.279 96.461 0.374 33.275 0.133 3.944 4.830 5.019 Λ = 0 0.519 93.334 0.518 47.434 0.099 7.196 12.026 12.026 Λ = 4 0.192 87.756 0.379 25.110 0.133 12.610 7.436 4.748 a = 10 0.192 97.557 0.377 27.929 0.132 2.847 4.964 5.482 a = 30 0.247 96.318 0.375 31.101 0.133 4.084 4.914 5.213 a = 70 0.297 96.650 0.373 34.544 0.133 3.756 4.776 4.898 a = 90 0.310 96.960 0.373 35.548 0.133 3.449 4.726 4.805 NC= 0.0% 0.277 96.351 0.379 33.132 0.133 4.058 4.838 5.081 NC= 0.2% 0.277 96.351 0.379 33.132 0.133 4.058 4.838 5.081 NC= 0.5% 0.275 96.182 0.397 32.919 0.135 4.244 4.842 5.311 NC= 1.0% 0.265 95.408 0.569 31.985 0.133 5.177 4.713 7.368 NS = 0.0% 0.335 99.304 0.067 38.132 0.222 0.847 3.383 1.527 NS = 0.2% 0.277 96.351 0.378 33.132 0.133 4.057 4.839 5.078 NS = 0.5% 0.266 95.470 0.553 32.057 0.132 5.099 4.733 7.167 NS = 1.0% 0.249 94.078 1.057 30.529 0.101 6.947 4.036 13.192 fN C = 0.0 0.284 96.782 0.376 33.699 0.131 3.626 5.553 4.892 fN C = 0.5 0.282 96.677 0.375 33.558 0.133 3.730 4.964 4.949 fN C = 1.0 0.275 96.182 0.374 32.919 0.131 4.220 4.817 5.089 K = 0.5V0(σ) 0.255 94.563 0.569 58.403 0.133 6.053 4.715 7.941 K = 0.8V0(σ) 0.270 95.834 0.397 41.641 0.135 4.602 4.855 5.480 K = 1.2V0(σ) 0.284 96.834 0.360 26.663 0.131 3.549 4.817 4.730 K = 1.5V0(σ) 0.291 97.234 0.363 19.414 0.595 3.233 4.600 4.913 N CW0= 0.2 0.261 95.091 0.369 31.623 0.131 5.301 7.445 4.875 N CW0= 0.5 0.297 97.607 0.381 34.886 0.133 2.810 2.988 5.235 N CW0= 1.0 0.317 98.634 0.378 36.644 0.600 1.957 1.274 6.623

(28)

3.2 The impact of put options on investment risk choices

When the firm value at maturity is lower than the strike price, the call option portion of the portfolio is worthless - the call option will not be exercised. When the manager is concerned about low firm values, will have a tendency to choose a higher risk level.

A severance package is the payment and benefits that employees receive when they leave employment at a company.

To approximate a severance package (explicit/implicit or the possibilty of them) in our model, we will include put options in the compensation portfolio.

In Table 3.3 we show the impact of put options in the investment risk choice.

Table 3.3: Number of regular calls and puts that performed the same manager’s utility as Table 3.1. σ V0(σ) VC 102NC 102NP TC 103 dE[U ] dV 10 3 _PPS Base 0.2719 95.952 32.633 0.331 1.179 4.473 4.734 4.930 Λ = 0 0.5492 90.085 47.551 0.345 0.208 10.389 12.122 12.122 Λ = 4 0.2152 90.591 27.307 0.150 1.633 9.586 11.866 2.124 a = 10 0.1797 97.221 27.052 0.364 12.890 3.230 4.872 5.920 a = 30 0.2367 95.735 30.250 0.357 2.347 4.714 4.819 5.510 a = 70 0.2891 95.991 33.788 0.350 0.988 4.463 4.670 5.061 a = 90 0.3039 96.391 34.922 0.349 0.824 4.066 4.609 4.924 NC= 0.2% 0.3117 98.386 36.178 0.150 0.326 1.816 4.204 2.381 NC= 0.5% 0.2680 95.654 32.273 0.365 1.377 4.812 4.732 5.402 NC= 1.0% 0.2617 95.156 31.695 0.420 1.750 5.377 4.706 6.188 NS= 0.0% 0.3461 99.602 39.003 0.082 0.130 0.512 3.408 1.409 NS= 0.2% 0.2875 97.037 34.051 0.242 0.696 3.279 4.622 3.673 NS= 0.5% 0.2641 95.346 31.912 0.400 1.604 5.162 4.718 5.897 NS= 1.0% 0.2641 95.346 31.912 0.400 1.604 5.162 4.717 5.897 fN C = 0.0 0.2766 96.295 33.061 0.332 1.104 4.133 5.416 4.789 fN C = 0.5 0.2758 96.239 32.990 0.332 1.116 4.189 4.848 4.848 fN C = 1.0 0.2688 95.714 32.345 0.331 1.234 4.708 4.725 5.000 K = 0.5V0(σ) 0.2227 91.428 56.137 0.528 9.535 9.349 4.564 10.122 K = 0.8V0(σ) 0.2570 94.764 40.562 0.335 1.954 5.688 4.804 5.430 K = 1.2V0(σ) 0.3078 98.196 28.907 0.200 0.365 2.058 4.414 2.916 K = 1.5V0(σ) 0.3547 99.778 25.562 0.100 0.095 0.347 3.501 1.500 N CW0 = 0.2 0.2523 94.357 30.821 0.329 1.606 6.054 7.349 4.812 N CW0 = 0.5 0.2938 97.424 34.609 0.334 0.889 3.017 2.906 5.124 N CW0 = 1.0 0.3609 99.874 40.136 0.200 0.282 0.406 0.975 3.555

(29)

of put options is choosen such that the put options’ market value is 10% of the call options. The number of calls is choosen in order to achieve the same utility level as Table 3.1.

Using put options is a good way to reduce the agency cost associated with deviations from the optimal risk level, not only because put options are also an increasing function of the volatility but also because a put insures the manager when the firm value is lower than expected. Therefore, the manager will be willing to take more risks even when the firm value is low.

However, when close to maturity date, if call options are deep out-of-the-money the manager may want to change her behaviour in order to have put options in-the-money.

This problem can be rectified if the manager may want to preserve her reputation, or with a constraint in the manager’s ability to change short-run stock price.

The severence packages have also been criticized because they are the last chance of a reward when the manager has poor performance.

However, in our results we can see that including put options can significantly reduce the total cost to the firm.

3.3 The roles of lookbacks calls in reducing managerial

incentive costs

As explained in the section 1.1.4, the payoff at maturity of a European lookback call option is the difference between the terminal firm value, denoted by VT, and the minimum firm value, Vmin

T . As this last value is defined as the minimum firm value, the following is noticed: VT − VTmin≥ 0.

In this section, we will explore the role of lookbacks in a manager’s portfolio and the incentive to reduce the managerial incentive cost.

(30)

Table 3.3 shows that put options have an opposite behaviour regarding this. However, they can provide the wrong incentives. Using lookback options instead of regular calls, Tables 3.4 and 3.5 are similar to Tables 3.1 and 3.2.

Table 3.4: Number of lookback calls that performed the same manager’s utility as Table 3.1.

σ V0(σ) VLB 102NL TC 103 dE[U ] dV 10 3 _PPS Base 0.272 95.952 45.783 0.331 4.516 4.734 4.930 Λ = 0 0.549 90.085 63.467 0.345 10.443 12.122 12.122 Λ = 4 0.215 90.591 37.851 0.150 9.602 11.866 2.124 a = 10 0.180 97.221 36.824 0.364 3.265 4.872 5.920 a = 30 0.237 95.735 42.206 0.357 4.756 4.819 5.510 a = 70 0.289 95.991 47.446 0.350 4.510 4.670 5.061 a = 90 0.304 96.391 49.037 0.349 4.115 4.609 4.924 NC= 0.2% 0.312 98.386 50.788 0.150 1.838 4.204 2.381 NC= 0.5% 0.268 95.654 45.263 0.365 4.859 4.732 5.402 NC= 1.0% 0.262 95.156 44.422 0.420 5.430 4.706 6.188 NS= 0.0% 0.346 99.602 54.596 0.082 0.525 3.408 1.409 NS= 0.2% 0.288 97.037 47.813 0.242 3.313 4.622 3.673 NS= 0.5% 0.264 95.346 44.739 0.400 5.213 4.718 5.897 NS= 1.0% 0.264 95.346 44.739 0.400 5.213 4.717 5.897 fN C = 0.0 0.277 96.295 46.401 0.332 4.177 5.416 4.789 fN C = 0.5 0.276 96.239 46.298 0.332 4.232 4.848 4.848 fN C = 1.0 0.269 95.714 45.368 0.331 4.751 4.725 5.000 K = 0.5V0(σ) 0.223 91.428 56.520 0.528 9.351 4.564 10.122 K = 0.8V0(σ) 0.257 94.764 46.296 0.335 5.707 4.804 5.430 K = 1.2V0(σ) 0.308 98.196 52.025 0.200 2.104 4.414 2.916 K = 1.5V0(σ) 0.355 99.778 62.692 0.100 0.384 3.501 1.500 N CW0 = 0.2 0.252 94.357 43.140 0.329 6.095 7.349 4.812 N CW0 = 0.5 0.294 97.424 48.602 0.334 3.063 2.906 5.124 N CW0 = 1.0 0.361 99.874 56.071 0.200 0.437 0.975 3.555

In Table 3.4, the number of lookbacks calls is chosen to yield the same utility level as each corresponding entry in Table 3.1.

As lookback options are always "in-the-money" the manager will want to take more risk and this will help with the agency costs associated with deviating from the optimal risk level. We can also see that the total cost is significantly lower when using lookbacks calls instead of regular calls.

Analysing the entries with different Λ’s, we can see that the incentive provided from lookbacks is always the better one. In the case where the manager is risk averse (Λ = 4), lookbacks calls induce more risk than regular calls. When the manager is risk neutral,

(31)

(Λ = 0), lookbacks induce less risk than regular options.

Table 3.5: Number of company shares and lookback calls that performed the same manager’s utility as Table 3.1. σ V0(σ) 102NS VLB 102NL TC 103 dE[U ] dV 10 3 _PPS Base 0.278 96.406 0.374 46.605 0.133 4.016 4.835 5.025 Λ = 0 0.519 93.334 0.516 63.876 0.098 7.210 11.972 11.972 Λ = 4 0.193 87.938 0.379 34.645 0.133 12.441 7.413 4.733 a = 10 0.194 97.598 0.377 38.493 0.132 2.821 4.959 5.476 a = 30 0.248 96.404 0.375 43.685 0.133 4.015 4.906 5.205 a = 70 0.297 96.650 0.373 48.511 0.133 3.775 4.776 4.898 a = 90 0.310 96.960 0.373 49.908 0.133 3.468 4.726 4.805 NC= 0.0% 0.277 96.351 0.379 46.503 0.133 4.075 4.838 5.081 NC= 0.2% 0.277 96.351 0.379 46.503 0.133 4.075 4.838 5.081 NC= 0.5% 0.275 96.182 0.397 46.196 0.135 4.261 4.569 5.012 NC= 1.0% 0.265 95.408 0.569 44.844 0.133 5.194 4.713 7.368 NS= 0.0% 0.335 99.304 0.067 53.442 0.222 0.881 3.383 1.527 NS= 0.2% 0.277 96.295 0.379 46.401 0.133 4.131 4.843 5.085 NS= 0.5% 0.266 95.470 0.553 44.949 0.132 5.116 4.733 7.167 NS= 1.0% 0.249 94.078 1.057 42.707 0.101 6.960 4.036 13.192 fN C = 0.0 0.284 96.782 0.375 47.314 0.133 3.644 5.551 4.891 fN C = 0.5 0.281 96.623 0.375 47.011 0.133 3.801 4.969 4.954 fN C = 1.0 0.275 96.182 0.373 46.196 0.133 4.238 4.816 5.088 K = 0.5V0(σ) 0.256 94.697 0.569 59.286 0.133 5.920 4.704 7.921 K = 0.8V0(σ) 0.270 95.834 0.397 47.915 0.135 4.611 4.855 5.478 K = 1.2V0(σ) 0.284 96.834 0.360 49.380 0.131 3.579 4.817 4.730 K = 1.5V0(σ) 0.286 96.936 0.379 58.742 0.600 3.780 4.627 5.125 N CW0 = 0.2 0.261 95.091 0.369 44.316 0.131 5.317 7.445 4.875 N CW0 = 0.5 0.297 97.607 0.381 48.991 0.133 2.829 2.988 5.235 N CW0 = 1.0 0.317 98.634 0.378 51.428 0.600 2.045 1.274 6.623

(32)

However, the same does not happen here - in the corresponding Table 3.5 lookback options are preferable. This happens because the manager will be wiling to take more risk as the lookback will never finish out of the money.

Continuing to compare Table 3.2 and corresponding Table 3.5 we can see that the total cost is even more reduced. Evaluated closely, Tables 3.4 and corresponding 3.5 show us that lookbacks are more effective than regular calls.

3.3.2 Interpreting lookbacks calls: option strike resetting

As we already noted above, lookback options comprove to be better incentives than restricted stocks or regular call options. In the following subsection we will tap into the pros and possible cons of this financial asset.

Looking at the results we can note that lookback call options can be a good incentive to the manager in her compensation package. Other argument for this is that they will not provide the wrong incentives to the manager when there are low stock prices, as the reset of the stock price in the lookback call is already taking in mind in the cost of the contract.

We can also take a look at the terminal stock price, VT. If this is positively correlated with any benchmark index, the lookback call strike price, V_Tmin, is also going to be correlated with that index. This makes an automatic mechanism which resets the option strike in a way that is consistent with the indexation. Therefore, this mechanism can help to filter out the actions of the manager - to analyse if her actions contributed to the decline of the stock price or if the factors to do so were beyond her control. This way, the manager will not be penalized for actions which factors will not depend on her neither be rewarded for stock price run-ups unrelated to her actions.

We will now take a look at a possible disavantage of the lookback calls. The manager may have the possibility of temporarily depress stock prices to a new low and then reverse that decision, creating a more valuable option because of the new lower strike price.

However, we argue that is much more difficult to increase value than decrease it, i.e., there is not a symmetry on those actions.

We can also note that for obtaining the same payoff, a higher return is needed when we start from a lower minimum. For example, suppose that we have a current and minimum price of $100. For a payoff of $20 we need a return of 20%. On the contrary, if we have a stock price of $50, we will need a return of 40% for the same payoff. It might be worthed to increase the price from $100 to $120, instead of drive the price from $100 to $50, and then back to $70.

(33)

the manager’s mind but they should be taking into account when using the strategy above. Other argument to not use the above is that their company stock components will decrease their value. We also remark that the advantages outweight this potential disadvantage.

(34)

4 Conclusions

One of the fairly variables that we should take in mind for the design of the managerial compensation structure is the pay-for-performance. To align the expectations of management positions with the shareholders the integration of stock options are an integral part of the process.

Depending on the managerial risk aversion, call options can induce over and under the optimal risk level for the firm. When a call option is deep out of the money they will not provide much incentive as the probability to finish in the money is low. However, a lookback option is always in the money and provides the incentive to increase the stock price above the current level.

Another interesting analysis is that lookbacks have similar characteristics to those em-bedded in indexed options because both terminal price and minimum price are presumably to be linked with market returns and their payoff filters out part of the market trend.

(35)

Bibliography

[1] Bekaert, G. & G. Wu. Asymmetric Volatility and Risk in Equity Markets. Review of

Financial Studies 13, pages 1–42, 2000.

[2] Black, F. & Scholes, M. The Pricing of Options and Corporate Liabilities. The Journal

of Political Economy, 81(3), pages 637–654, 1973.

[3] Boyle, P. & Y. Tian. Pricing Lookback and Barrier Options under the CEV Process.

Journal of Financial and Quantitative Analysis 34, pages 241–264, 1999.

[4] Cox, J. Notes on Option Pricing I: Constant Elasticity of Variance Diffusions.

Work-ing paper, Stanford University (reprinted in Journal of Portfolio Management, 1996, December, 15-17), page 15, 1975.

[5] Dennis, P. & S. Mayhew. Risk-Neutral Skewness: Evidence from Stock Options. Journal

of Financial and Quantitative Analysis 37, pages 471–493, 2002.

[6] Domingues, A. The valuation of turbo warrants under the CEV model. Master’s Thesis,

ISCTE and FCUL, 2012.

[7] Goldman, M. & Sosin, B. & Gatto, M. Path-dependent options: buy at the low, sell at the high. J. Financ. 34, pages 1111–1127, 1979.

(36)

compen-[11] Parrino, R. & Weisbach, M. Measuring investment distortions arising from stockholder-bondholder conflicts. J. Financ. Econ. 53, pages 3–42, 1999.

[12] Ross, S. Compensation, incentives, and the duality of risk aversion and riskiness. J.

Financ. 59, pages 207–225, 2004.

[13] Zhang, P. Exotic Options: A Guide to Second Generation Options, second ed. World

Options in managerial compensation